The concept of unbounded order convergence or \(uo\) -convergence was introduced in [1]} and is proposed firstly in [2]}. It has recently been intensively studied in several papers [3]}, [4]}, [5]}. Recall that a net \((x_\alpha )_{\alpha \in \mathcal {A}}\)  in a Riesz space \(E\)  is order convergent (or, \(o\) -convergent for short) to \(x\in E\) , denoted by \(x _\alpha \xrightarrow{}x\)  whenever there exists another net \((y_\beta )_{\beta \in \mathcal {B}}\)  in \(E\)  such that \(y_\beta \downarrow 0\)  and that for every \(\beta \in \mathcal {B}\) , there exists \(\alpha _0\in \mathcal {A}\)  such that \(|x_\alpha -x| \le y_\beta \)  for all \(\alpha \ge \alpha _0\) . A net \((x_\alpha )\)  in a Riesz space \(E\)  is unbounded order convergent (or, \(uo\) -convergent for short) to \(x\in E\)  if \(|x_\alpha -x|\wedge u\xrightarrow{}0\)  for all \(u\in E^+\) . We denote this convergence by \(x _\alpha \xrightarrow{}x\)  and write that \(x_\alpha \)  \(uo\) -convergent to \(x\) . This is an analogue of pointwise convergence in function spaces. Let \(\mathbb {R}^A\)  be the Riesz space of all real-valued functions on a non-empty set \(A\) , equipped with the pointwise order. It is easily seen that a net \((x_\alpha )\)  in \(\mathbb {R}^A\)  \(uo\) -converges to \(x\in \mathbb {R}^A\)  if and only if it converges pointwise to \(x\) . For instance in \(c_0\)  and \(\ell _p\) (\(1\le p\le \infty )\) , \(uo\) -convergence of nets is the same as coordinate-wise convergence. Assume that \((\Omega ,\Sigma ,\mu )\)  is a measure space and let \(E=L_p(\mu )\)  for some \(1\le p<\infty \) . Then \(uo\) -convergence of sequences in \(L_p(\mu )\)  is the same as almost everywhere convergence. In [6]}, Wickstead characterized the spaces in which weak convergence of nets implies \(uo\) -convergence and vice versa. In [3]}, Gao characterized the space \(E\)  such that in its dual space \(E^*\) , \(uo\) -convergence implies \(w^*\) -convergence and vice versa. He also characterized the spaces in whose dual space simultaneous \(uo\) - and \(w^*\) -convergence imply weak/norm convergence. We show that the collection of all order bounded strongly order continuous linear functionals on a Riesz space \(E\)  is a band of \(E^\sim \)  [Theorem 2.7]. For unexplained terminology and facts on Banach lattices and positive operators, we refer the reader to [8]}, [9]}. Let us start with the definition.
